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Embedding metric spaces into normed spaces and estimates of metric capacity. (English) Zbl 1127.52006

Summary: Let \({\mathcal M}^d\) be an arbitrary real normed space of finite dimension \(d \geq 2\). We define the metric capacity of \({\mathcal M}^d\) as the maximal \(m \in {\mathbb N}\) such that every \(m\)-point metric space is isometric to some subset of \({\mathcal M}^d\) (with metric induced by \({\mathcal M}^d\)). We obtain that the metric capacity of \({\mathcal M}^d\) lies in the range from 3 to \(\left\lfloor\frac{3}{2}d\right\rfloor+1\), where the lower bound is sharp for all \(d\), and the upper bound is shown to be sharp for \(d \in \{2, 3\}\). Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from \(d + 2\) to \(\left\lfloor\frac{3}{2}d\right\rfloor+1\).

MSC:

52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52B10 Three-dimensional polytopes
52C10 Erdős problems and related topics of discrete geometry
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